How can i PROVE to my wife i am changing during separation?

The only way to prove it is to show her with your actions. Unfortunately, words are just words, you can say you have changed all you want, but the only way she will know is if you follow these words up with your actions. To me, it doesn't sound like she is leaving for good and when she comes back you will have the opportunity to show her. Just tell her that you know words are just words and you are going to show her in your day-to-day actions that you have changed. Then do just that. Good luck!